On the spectrum of narrow Neumann waveguide with periodically distributed ${\delta }^{\prime }$ traps

Abstract

We analyze a family of singular Schrödinger operators describing a Neumann waveguide with a periodic array of singular traps of a ${\delta }^{\prime }$ type. We show that in the limit when perpendicular size of the guide tends to zero and the ${\delta }^{\prime }$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.

14 pages